Chapter 12: Q10P (page 518)
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12.48: An electromagnetic plane wave of (angular) frequency is travelling in the direction through the vacuum. It is polarized in the direction, and the amplitude of the electric field is .
(a) Write down the electric and magnetic fields, role="math" localid="1658134257504" and [Be sure to define any auxiliary quantities you introduce, in terms of , , and the constants of nature.]
(b) This same wave is observed from an inertial system moving in thedirection with speed relative to the original system . Find the electric and magnetic fields in , and express them in terms of the role="math" localid="1658134499928" coordinates: and . [Again, be sure to define any auxiliary quantities you introduce.]
(c) What is the frequency of the wave in ? Interpret this result. What is the wavelength of the wave in ? From and , determine the speed of the waves in . Is it what you expected?
(d) What is the ratio of the intensity in to the intensity in? As a youth, Einstein wondered what an electromagnetic wave would like if you could run along beside it at the speed of light. What can you tell him about the amplitude, frequency, and intensity of the wave, as approaches ?
Show that the potential representation (Eq. 12.133) automatically satisfies 
[Suggestion: Use Prob. 12.54.]
Show that it is possible to outrun a light ray, if you're given a sufficient head start, and your feet generate a constant force.
(a) Equation 12.40 defines proper velocity in terms of ordinary velocity. Invert that equation to get the formula for u in terms of .
(b) What is the relation between proper velocity and rapidity (Eq. 12.34)? Assume the velocity is along the x direction, and find as a function of .
Question: A stationary magnetic dipole, , is situated above an infinite uniform surface current, (Fig. 12.44).
(a) Find the torque on the dipole, using Eq. 6.1.
(b) Suppose that the surface current consists of a uniform surface charge , moving at velocity , so that , and the magnetic dipole consists of a uniform line charge , circulating at speed (same ) around a square loop of side I , as shown, so that .Examine the same configuration from the point of view of system, moving in the direction at speed . In , the surface charge is at rest, so it generates no magnetic field. Show that in this frame the current loop carries an electric dipole moment, and calculate the resulting torque, using Eq. 4.4.
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